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Find f. f "(x) = x-2, x > 0, f(1) = 0, F(3) = 0 %3D %3D In (1) |-In(\x|) + In(1)x + f(x)

Find f. f "(x) = x-2, x > 0, f(1) = 0, F(3) = 0 %3D %3D In (1) |-In(\x|) + In(1)x + f(x)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement** Find the function \( f \). Given: \[ f''(x) = x^{-2}, \quad x > 0 \] \[ f(1) = 0, \quad f(3) = 0 \] **Solution:** The function is determined as: \[ f(x) = -\ln(|x|) + \frac{\ln(1)}{2}x + \frac{\ln(1)}{2} \] **Explanation:** 1. **Differential Equation**: The problem involves solving a second-order differential equation where the second derivative of \( f \) with respect to \( x \) is given as \( x^{-2} \). 2. **Initial Conditions**: The function \( f \) is subject to initial conditions where \( f(1) = 0 \) and \( f(3) = 0 \). These conditions help in determining the constants of integration after solving the differential equation. 3. **Function Expression**: The function \( f(x) \) contains a natural logarithm, an absolute value, and terms involving logarithm-based constants. The expression uses the properties of logarithms and the absolute value function. This solution involves calculus, particularly integration, to find \( f(x) \) that satisfies both the differential equation and the initial conditions provided.
Transcribed Image Text:**Problem Statement** Find the function \( f \). Given: \[ f''(x) = x^{-2}, \quad x > 0 \] \[ f(1) = 0, \quad f(3) = 0 \] **Solution:** The function is determined as: \[ f(x) = -\ln(|x|) + \frac{\ln(1)}{2}x + \frac{\ln(1)}{2} \] **Explanation:** 1. **Differential Equation**: The problem involves solving a second-order differential equation where the second derivative of \( f \) with respect to \( x \) is given as \( x^{-2} \). 2. **Initial Conditions**: The function \( f \) is subject to initial conditions where \( f(1) = 0 \) and \( f(3) = 0 \). These conditions help in determining the constants of integration after solving the differential equation. 3. **Function Expression**: The function \( f(x) \) contains a natural logarithm, an absolute value, and terms involving logarithm-based constants. The expression uses the properties of logarithms and the absolute value function. This solution involves calculus, particularly integration, to find \( f(x) \) that satisfies both the differential equation and the initial conditions provided.
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